Polarized relations at singulars over successors

Abstract

Erdos, Hajnal and Rado asked whether $\binom{\aleph_{\omega_1}}{\aleph_2}\rightarrow\binom{\aleph_{\omega_1}}{\aleph_0}_2$ and whether $\binom{\aleph_{\omega_1}}{\aleph_2}\rightarrow\binom{\aleph_{\omega_1}}{\aleph_1}_2$. We prove that both relations are independent over ZFC. We shall also prove that $\binom{\mu}{\aleph_2}\rightarrow\binom{\mu}{\aleph_2}_2$ is independent over ZF for some $\mu$ of cofinality $\omega_1$.